Let $H_n=\sum_{k=1}^n\frac 1 k$ be the $n$-th harmonic number with $H_0=0.$

**Question:** Is the following true?
$$\det\left(H_{i+j}\right)_{i,j=0}^n=(-1)^n \frac{2H_{n}}{n! \prod_{j=1}^n \binom{2j}{j} \binom{2j-1}{j}}.$$

**Edit:**
Comparing with the orthogonal polynomials whose moments are the numbers $\frac{1}{n+1}$ it suffices to show the following identity:
$$\sum_{j=0}^n (-1)^j\frac{\binom n j \binom{n+j} j}{\binom{2n} n} H_j \prod_{j=0}^{n-1}\frac{(j!)^3}{(n+j)!} = (-1)^n \frac{2H_n}{n! \prod_{j=1}^n \binom{2j}{j} \binom{2j-1}{j}}.$$